Exact Demonstration of pair-density-wave superconductivity in the $\sigma_z$-Hubbard model

Describing and achieving `unconventional' superconductivity remains a forefront challenge in quantum many-body physics. Here we use a unitary mapping, combined with the well-established properties of the attractive Hubbard model to demonstrate rigorously a Hamiltonian with a low temperature pair-density-wave (PDW) phase. We also show that the same mapping, when applied to the widely accepted properties of the repulsive Hubbard model, leads to a Hamiltonian exhibiting triplet $d$-wave PDW superconductivity and an unusual combination of ferro- and antiferro-magnetic spin correlations. We then demonstrate the persistence of the $d$-wave PDW in a Hamiltonian derived from the mapping of the extended $t$-$J$ model in the large-$U$ limit. Furthermore, through strategic manipulation of the nearest-neighbor hopping signs of spin-down electrons, we illustrate the attainability of PDW superconductivity at other momenta. The intertwining of different magnetic and exotic pairing correlations noted here may have connections to experimental observations in spin-triplet candidates like UTe$_2$.